A hypergraph H = (V, E) is said to be a Hausdorff hypergraph if for any two distinct vertices u, v of V there exist hyperedges e1, e2 ∈ E such that u ∈ e1, v ∈ e2 and e1 ∩ e2 = ∅. In this paper we have discussed hausdorff property of hypergraphs as well as minimal hausdorff hypergraph. Previous work on Hausdorff-type separation properties, driven by classical topology and graph theory, has included the study of specific types of hypergraphs to address vertex separability with disjoint hyperedges. Continuing from this stream of research, this work aims to conduct an in-depth study of Hausdorff hypergraphs, with special emphasis on minimal Hausdorff hypergraphs and their variants. We obtain results on minimal Hausdorff hypergraphs with respect to bounds of the number of hyperedges and study sufficient conditions for competition hypergraphs of digraphs and independent hypergraphs of graphs to be Hausdorff. Links with conformal hypergraphs, cyclomatic number, and acyclicity are considered in an attempt to cover

Distance-based (and path-based) covering problems for graphs of given cyclomatic number

Let G be a connected graph with vertex set V ( G ) and edge set E ( G ) . The Laplacian matrix of G is defined as L ( G ) = D ( G ) − A ( G ) , where D ( G ) is a diagonal matrix of degrees of the vertices of G and A ( G ) is the adjacency matrix of G. The multiplicity of an eigenvalue μ of L ( G ) is denoted by m G ( μ ) . In 2022, Wen et al. [Czech. Math. J., 72(2022)] proved that, if G is not a cycle, then m G ( μ ) ≤ 2 c ( G ) + p ( G ) − 1 , where c ( G ) = | E ( G ) | − | V ( G ) | + 1 is the cyclomatic number of G and p ( G ) is the number of pendant vertices of G. We characterize the graph G with m G ( 1 ) = 2 c ( G ) + p ( G ) − 1 .

Degree-based function index of graphs with given bipartition and small cyclomatic number

Road structure determines road network connectivity and enhances the level of accessibility. Therefore, the study evaluates road network connectivity and accessibility among small-sized urban centres in Kogi State, Nigeria. The study adopted purposive sampling techniques to select (20) local government headquarters in Kogi State which are referred to as the (SSUCs). Data on the existing road network, edges, vertices and density was obtained from the digitised road network map of the study area. The method of analysis includes beta, alpha, eta and gamma indices including cyclomatic number, aggregate transport score and road density. Finding of the study shows a total of 918 vertices and 770 edges. The calculated total road length (TRL) 851.43, Land area (LA) 196.93sqkm, Alpha index (AI), -1.63, Beta Index (BI), 16.08 and Gamma Index (GI) 5.67. Also, cyclomatic index (CI), -16.8, Aggregate transport score (ATS),-147.58, Eta Index (EI), 25.88 and Road density (RD) 101.99. The study concludes that the SSUCs in Kogi State have poor road network connectivity and this affects the level of accessibility. Therefore, the study recommends the construction of more road networks by the government to enhance accessibility among the SSUCs in the state. Also, Kogi state government should put in place a road infrastructure development plan and policy for implementation.

Abstract For a finite graph, we establish natural isomorphisms between eigenspaces of a Laplace operator acting on functions on the edges and eigenspaces of a transfer operator acting on functions on one-sided infinite non-backtracking paths. Interpreting the transfer operator as a classical dynamical system and the Laplace operator as its quantization, this result can be viewed as a quantum-classical correspondence. In contrast to previously established quantum-classical correspondences for the vertex Laplacian which exclude certain exceptional spectral parameters, our correspondence is valid for all parameters. This allows us to relate certain spectral quantities to topological properties of the graph such as the cyclomatic number and the 2-colorability. The quantum-classical correspondence for the edge Laplacian is induced by an edge Poisson transform on the universal covering of the graph which is a tree of bounded degree. In the special case of regular trees, we relate both the vertex and the edge Poisson transform to the representation theory of the automorphism group of the tree and study associated operator valued Hecke algebras.

The paper presents the results of the analysis and synthesis of bucket elevator drives based on combined graph models in mechanics. The study is carried out using the theory of modified kinematic graphs, which makes it possible to represent bucket elevator drives as elastic mechanical systems with mechanical feedback. The proposed approach provides a formalized analysis of the structures of existing drives and creates a basis for the synthesis of new design solutions with extended functional capabilities. A review and systematization of the main types of graph models used to solve problems in mechanics is performed, including kinematic, force, and combined graphs. The expediency of using modified kinematic graphs for the analysis of controlled elastic systems characterized by negative or zero mobility is demonstrated. The degree of mobility and the cyclomatic number are used to assess the controllability of elastic characteristics of mechanisms. The paper proposes a criterion for selecting an optimal design solution based on the energy of a modified kinematic graph, which is determined using spectral graph theory. Adjacency matrices are formed taking into account weighting coefficients that reflect design and technological factors as well as the direction of elastic force action. Using the example of two bucket elevator drive mounting structures, a comparative analysis is performed, which substantiates the selection of a more rational structure according to the criterion of minimum graph energy. The obtained results confirm the effectiveness of using modified kinematic graphs for the formalization and algorithmization of structural analysis and synthesis processes of bucket elevator drives, taking into account design and technological constraints.

For a graph G with edge set E, let d(u) denote the degree of a vertex u in G.The diminished Sombor (DSO) index of G is defined as. The cyclomatic number of a graph is the smallest number of edges whose removal makes the graph acyclic.A connected graph of maximum

Multiplicity of signless Laplacian eigenvalue 2 of a connected graph with a perfect matching

The eigenvalue multiplicity of line graphs

Bounds of nullity for complex unit gain graphs

We consider the adjacency spectrum of cycle-spliced signed graphs (CSSG), i.e., signed graphs whose blocks are (independent) signed cycles. For a signed graph Σ, the nullity η(Σ) is the multiplicity of the 0-eigenvalue. The adjancency spectrum of cycle-spliced (signed) graphs is studied in the literature for the relation between the nullity η and the cyclomatic number c, in particular, it is known that 0≤η(Σ) ≤ c(Σ)+1. In this paper, nonsingular cycle-spliced bipartite signed graphs are characterized. For cycle-spliced signed graphs Σ having only odd cycles, we show that η(Σ) is 0 or 1. Finally, we compute the nullity of CSSGs consisting of at most three cycles.

The study aimed to obtain relationships between the omega invariants of a graph and its complement. We used some graph parameters, including the cyclomatic numbers, number of components, maximum number of components, order, and size of both graphs G and G. Also, we used triangular numbers to obtain our results related to the cyclomatic numbers and omega invariants of G and G. Several bounds for the above graph parameters have been obtained by the direct application of the omega invariant. We used combinatorial and graph theoretical methods to study formulae, relations, and bounds on the omega invariant, the number of faces, and the number of compo-nents of all realizations of a given degree sequence. Especially so-called Nordhaus-Gaddum type resulted in our calculations. In these calculations, the triangular numbers less than a given number play an important role. Quadratic equations and inequalities are intensively used. Several relations between the size and order of the graph have been utilized in this study. In this paper, we have obtained relationships between the omega invariants of a graph and its complement in terms of several graph parameters, such as the cyclomatic numbers, number of components, maximum number of components, order, and size of G and G, and triangular numbers. Some relationships between the omega invariants of a graph and its complement have been obtained.

In this work, we studied the problem of determining the values of the Zagreb indices of all the realizations of a given degree sequence. We first obtained some new relations between the first and second Zagreb indices and the forgotten index sometimes called the third Zagreb index. These relations also include the triangular numbers, order, size, and the biggest vertex degree of a given graph. As the first Zagreb index and the forgotten index of all the realizations of a given degree sequence are fixed, we concentrated on the values of the second Zagreb index and studied several properties including the effect of vertex addition. In our calculations, we make use of a new graph invariant, called omega invariant, to reach numerical and topological values claimed in the theorems. This invariant is closely related to Euler characteristic and the cyclomatic number of graphs. Therefore this invariant is used in the calculation of some parameters of the molecular structure under review in terms of vertex degrees, eccentricity, and distance.

On the graphs of a fixed cyclomatic number and order with minimum general sum-connectivity and Platt indices

The net Laplacian matrix of a signed graph \(\Gamma = (G, \sigma)\), where \(G = (V(G),E(G))\) is an unsigned graph (referred to as the underlying graph) and \(\sigma: E(G) \rightarrow \{-1, +1\}\) is the sign function, is defined as \(L^{\pm}(\Gamma) = D^{\pm}(\Gamma) - A(\Gamma)\). Here, \(D^{\pm}(\Gamma)\) and \(A(\Gamma)\) represent the diagonal matrix of net-degrees and the adjacency matrix of \(\Gamma\), respectively. The nullity of \(L^{\pm}(\Gamma)\), denoted as \(\eta (L^{\pm} (\Gamma))\), refers to the multiplicity of \(0\) as an eigenvalue of \(L^{\pm}(\Gamma)\). In this paper, we concentrate on the nullity of the net Laplacian matrix of a connected signed graph \(\Gamma\), and establish that \(1 \leq \eta (L^{\pm} (\Gamma)) \leq \min\{ \beta(\Gamma) + 1, |V(\Gamma)| - 1 \}\), where \(\beta(\Gamma) = |E(\Gamma)| - |V(\Gamma)| + 1\) denotes the cyclomatic number of \(\Gamma\). We completely determine the connected signed graphs with nullity \(|V(\Gamma)| - 1\). Additionally, we characterize the signed cactus graphs with nullity \(1\) or \(\beta(\Gamma) + 1\).

Eigenvalue multiplicity of graphs with given cyclomatic number and given number of quasi-pendant vertices

A characterization of trees with eigenvalue multiplicity one less than their number of pendant vertices

Abstract Let G = (V, E) be a simple graph with n vertices and m edges. ν(G) and c(G) = m − n + θ be the matching number and cyclomatic number of G, where θ is the number of connected components of G, respectively. Wang and Wong in [18] provided formulae for the upper and the lower bounds of the nullity η(G) of G as η(G) = n − 2ν(G) + 2c(G) and η(G) = n − 2ν(G) − c(G), respectively. In this paper, we restate the upper and the lower bounds of nullity η(G) of G utilizing omega invariant and inherently vertex degrees of G. Also, in the case of the maximal and the minimal nullity conditions are satisfied for G, we present both two main inequalities and many inequalities in terms of Omega invariant, analogously cyclomatic number, number of connected components and vertex degrees of G.

Lower bounds on the general first Zagreb index of graphs with low cyclomatic number